nLab
model structure on functors

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Contents

Idea

The model category structures on functor categories are models for (∞,1)-categories of (∞,1)-functors.

For CC a model category and DD any small category there are two “obvious” ways to put a model category structure on the functor category [D,C][D,C], called the projective and the injective model structures. For completely general CC, neither one need exist. The projective model structure exists as long as CC is cofibrantly generated, while the injective model structure exists as long as CC is combinatorial.

A related kind of model structure is the Reedy model structure/generalized Reedy model structure on functor categories, which applies for any model category CC, but requires DD to be a very special sort of category, namely a Reedy category/generalized Reedy category.

In the special case that C=C = SSet is the standard model category of simplicial sets the projective and injective model structure on the functor categories [D,SSet][D,SSet] are described in more detail at global model structure on simplicial presheaves.

Definition

Definition

For CC a combinatorial model category or, in the projective case, just a cofibrantly generated model category, and DD a small category there exist the following two (combinatorial) model category structures on the functor category [D,C][D,C]:

  • the projective structure [D,C] proj[D,C]_{proj}: weak equivalences and fibrations are the natural transformations that are objectwise such morphisms in CC.

  • the injective structure [D,C] inj[D,C]_{inj}: weak equivalences and cofibrations are the natural transformations that are objectwise such morphisms in CC.

More generally, if CC is in addition a simplicial model category and DD a smooth sSet-enriched category, then the sSet-enriched functor category, also denoted [D,C][D,C], carries the above two model strutures.

Properties

In all of the following, let S\mathbf{S} be an excellent model category. The standard example is the model structure on simplicial sets, sSet QuillensSet_{Quillen}.

Let DD (and D 1D_1, D 2D_2, …) be a combinatorial S\mathbf{S}-enriched model category.

Moreover, CC in the following is assumed to be either an ordinary small category, or, more generally, it is a small S\mathbf{S}-enriched category.

If S=\mathbf{S} = sSet Quillen{}_{Quillen} and CC is an ordinary small category, then then model structures discussed here are instances of the model structure on simplicial presheaves. If CC is itself sSetsSet-enriched, then they are instances of the model structure on sSet-enriched presheaves.

General

Proposition

The projective and injective structures [D,C] proj[D,C]_{proj} and [D,C] inj[D,C]_{inj}, def. 1

The existence of the unenriched model structure apears as HTT, prop. A.2.8.2 The enriched case is HTT, prop. A.3.3.2 and the remarks following that. The statement about properness appears as HTT, remark A.2.8.4.

Cofibrant generation

Proposition

Let CC be an ordinary small category.

The cofibrations in [C,A] proj[C, A]_{proj} are generated from (i.e. are the weakly saturated class of morphisms defined by) the morphisms of the form

Id C(c,)i:C(c,)aC(c,)b Id_{C(c,-)}\cdot i : C(c,-)\cdot a \to C(c,-) \cdot b

for all cCc \in C and i:abi : a \to b a generating cofibration in AA. Here the dot denotes the tensoring of AA over sets, i.e. C(c,)aC(c,-)\cdot a is the functor that sends cCc' \in C to the coproduct C(c,c)A\coprod_{C(c,c')} A of |C(c,c)||C(c,c')| copies of AA.

In particular, every cofibration if [C,A] proj[C,A]_{proj} is in particular a cofibration in [C,A] inj[C,A]_{inj}. Similarly, every fibration in [C,A] inj[C,A]_{inj} is in particular a fibration in [C,A] proj[C,A]_{proj}

This is argued in the beginning of the proof of HTT, lemma A.2.8.3.

Relation to other model structures

Corollary

The identity functors

[D,C] injIdId[D,C] proj [D,C]_{inj} \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}} [D,C]_{proj}

form a Quillen equivalence (with Id:[D,C] proj[D,C] injId : [D,C]_{proj} \to [D,C]_{inj} being the left Quillen functor).

If DD is a Reedy category this factors through the Reedy model structure

[D,C] injIdId[D,C] ReedyIdId[D,C] proj [D,C]_{inj} \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}} [D,C]_{Reedy} \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}} [D,C]_{proj}

Functoriality in domain and codomain

Proposition

The functor model structures depend Quillen-functorially on their codomain, in that for

D 1RLD 2 D_1 \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D_2

a S\mathbf{S}-enriched Quillen adjunction between combinatorial S\mathbf{S}-enriched model categories, postcomposition induces S\mathbf{S}-enriched Quillen adjunctions

[C,D 1] proj[C,R][C,L][C,D 2] proj [C,D_1]_{proj} \stackrel{\overset{[C,L]}{\leftarrow}}{\underset{[C,R]}{\to}} [C,D_2]_{proj}

and

[C,D 1] inj[C,R][C,L][C,D 2] inj. [C,D_1]_{inj} \stackrel{\overset{[C,L]}{\leftarrow}}{\underset{[C,R]}{\to}} [C,D_2]_{inj} \,.

Moreover, if (LR)(L \dashv R) is a Quillen equivalence, then so is ([C,L][C,R])([C,L] \dashv [C,R]).

For the case that CC is a small category this is (Lurie, remark A.2.8.6), for the enriched case this is (Lurie, prop. A.3.3.6).

The Quillen-functoriality on the domain is more asymmetric.

Proposition

For p:C 1C 2p : C_1 \to C_2 a functor between small categories or an S\mathbf{S}-enriched functor between S\mathbf{S}-enriched categories, let

(p !p *p *):[C 2,D]p *p *p ![C 1,D] (p_! \dashv p^* \dashv p_*) : [C_2,D] \stackrel{\overset{p_!}{\to}}{\stackrel{\overset{p^*}{\to}}{\underset{p_*}{\leftarrow}}} [C_1,D]

be the adjoint triple where p *p^* is precomposition with pp and where p !p_! and p *p_* are left and right Kan extension along pp, respectively.

Then we have Quillen adjunctions

(p !p *):[C 1,D] projp *p ![C 2,D] proj (p_! \dashv p^*) : [C_1,D]_{proj} \stackrel{\overset{p_!}{\to}}{\underset{p^*}{\leftarrow}} [C_2,D]_{proj}

and

(p *p *):[C 1,D] injp *p *[C 2,D] inj. (p^* \dashv p_*) : [C_1,D]_{inj} \stackrel{\overset{p^*}{\leftarrow}}{\underset{p_*}{\to}} [C_2,D]_{inj} \,.

For CC not enriched this appears as (Lurie, prop. A.2.8.7), for the enriched case it appears as (Lurie, prop. A.3.3.7).

Remark

In the sSetsSet-enriched case, if p:D 1D 2p : D_1 \to D_2 is an weak equivalence in the model structure on sSet-categories, then these two Quillen adjunctions are both Quillen equivalences.

Proposition

For CC a combinatorial simplicial model category, the (∞,1)-category presented by [D,C] proj[D,C]_{proj} and [D,C] inj[D,C]_{inj} under the above assumptions is the (∞,1)-category of (∞,1)-functors Func(D,C )Func(D,C^\circ) from the ordinary category DD to the (,1)(\infty,1)-category presented by CC.

See (∞,1)-category of (∞,1)-functors for details.

Relation to homotopy Kan extensions/limits/colimits

Often functors DCD \to C are thought of as diagrams in the model category CC, and one is interested in obtaining their homotopy limit or homotopy colimit or, generally, for p:DDp : D \to D' any functor, their left and right homotopy Kan extension.

These are the left and right derived functors HoLan:=𝕃p 1HoLan := \mathbb{L} p_1 and HoRan:=p *HoRan := \mathbb{R} p_* of

[D,C] projp ![D,C] proj [D,C]_{proj} \stackrel{p_!}{\to} [D',C]_{proj}

and

[D,C] injp *[D,C] inj [D,C]_{inj} \stackrel{p_*}{\to} [D',C]_{inj}

respectively.

For more on this see homotopy Kan extension. For the case that D=*D' = * this reduces to homotopy limit and homotopy colimit.

Examples

Examples of cofibrant objects in the projective model structure are discussed at

References

The plain situation is the topic of section A.2.8 of

The enriched situation is section A.3.3 there.

See also

Revised on November 10, 2014 20:11:17 by Anonymous Coward (136.152.208.147)